The New Sudoku Players' Forum

[m_b_metcalf]

A

Code: Select all

 . . . . . . . 2 3
 . 1 . . . . 4 . .
 . . . 1 . . 7 8 .
 1 . . 2 7 5 . . .
 . . . . . 6 . 1 .
 . . . 9 . 4 . . .
 2 . . . 8 . . 7 .
 . . 6 . . . . . 5
 8 4 . . . 9 . . .  23C

[m_b_metcalf]

B

Code: Select all

+---+---+---+
|...|...|..9|
|.1.|...|4..|
|...|1..|8.3|
+---+---+---+
|1..|...|6..|
|...|71.|98.|
|...|...|24.|
+---+---+---+
|92.|.6.|...|
|.45|.92|...|
|3..|..4|...|
+---+---+---+   23 clues .... but cant get rid of the interlopping clue !


I suspect that B is either impossible or is extremly rare (I found no valid puzzle in 500,000 tries).

Mike

[JPF]

Code: Select all

A
+---+---+---+
|...|...|.6.|
|.1.|...|...|
|...|1..|983|
+---+---+---+
|1..|.3.|...|
|...|5.6|1..|
|...|.7.|...|
+---+---+---+
|6.9|..2|4..|
|..3|...|..8|
|.2.|.5.|7..|
+---+---+---+   21C


I suspect that B is either impossible or is extremly rare

I agree.

JPF

[coloin]

Therefore ..... In the style of JPF

Code: Select all

+---+---+---+
|1..|...|234|
|...|...|561|
|...|1..|789|
+---+---+---+
|..1|...|893|
|...|...|172|
|...|...|456|
+---+---+---+
|425|368|917|
|768|291|345|
|319|754|628|
+---+---+---+    43 sols


[m_b_metcalf]

B

Code: Select all

 . . . . . . 1 9 7
 . 1 . . . . 3 8 4
 . . . 1 . . 6 2 5
 1 . . . . . 4 6 2
 . . . . 1 . 5 7 9
 . . . . . . 8 1 3
 6 8 4 2 3 7 9 5 1
 2 3 1 6 9 5 7 4 8
 5 7 9 4 8 1 2 3 6   4 solutions

[JPF]

Code: Select all

A
+---+---+---+
|...|...|.9.|
|.1.|...|...|
|...|1..|582|
+---+---+---+
|1..|...|...|
|...|263|1..|
|...|.7.|...|
+---+---+---+
|8.9|..4|...|
|..3|..7|.5.|
|.2.|...|7..|
+---+---+---+    20C


JPF

[coloin]

I suspect that B is either impossible or is extremly rare (I found no valid puzzle in 500,000 tries).Mike

yes I think its impossible as its unlikely with r3c4 = r4c3 it will ever be valid

Code: Select all

+---+---+---+
|...|...|123|
|...|...|456|
|...|1..|789|
+---+---+---+
|..1|...|932|
|...|...|547|
|...|...|861|
+---+---+---+
|629|875|314|
|453|961|278|
|178|243|695|
+---+---+---+    best was 7 solutions !


the 1@r2c2 and the1@r5c5 are placed

[m_b_metcalf]

B

Code: Select all

 . . . . . . 3 9 1
 . 1 . . . . 2 8 6
 . . . 1 . . 4 5 7
 1 . . . . . 5 3 4
 . . . . 1 . 6 2 9
 . . . . . . 1 7 8
 7 9 3 5 4 6 8 1 2
 5 8 6 7 2 1 9 4 3
 4 2 1 3 8 9 7 6 5  5 solutions

These low solution counts (<6) occur at a rate of less than one per million.

[JPF]

B

Code: Select all

+---+---+---+
|...|...|197|
|.1.|...|384|
|...|1..|625|
+---+---+---+
|1..|...|462|
|...|.1.|579|
|...|...|813|
+---+---+---+
|684|237|951|
|231|695|748|
|579|481|236|
+---+---+---+   4 solutions



Unless we're mistaken, I agree with Mike: the minimum number of solutions is 4.
The corollary question is now: How many cells can be placed?
In Mike's example, the number of placed cells is 16 (65-49)

Code: Select all

+---+---+---+
|..5|.4.|197|
|71.|5.9|384|
|94.|17.|625|
+---+---+---+
|19.|75.|462|
|...|.14|579|
|457|9..|813|
+---+---+---+
|684|237|951|
|231|695|748|
|579|481|236|
+---+---+---+


What are the lowest and the greatest number of cells that can be placed?

JPF

[coloin]

Unless we're mistaken...

No you are not mistaken ... it looks like all 4 boxes that are solved with 2 clues have different 2 clues
if there was one [4box] which solved with 2 siimilar clues there would be 10000 or more examples ... eg a random B1247 has many solutions

Code: Select all

+---+---+---+
|123|456|789|
|784|291|...|
|659|387|...|
+---+---+---+
|462|175|...|
|391|824|...|
|578|963|...|
+---+---+---+
|2..|...|...|
|9..|...|...|
|8..|...|...|
+---+---+---+  over 10000 solutions


In terms of crossing gangsters ..... dukuso posted that "there are essentially 2802 different [B1]B2B3B4B7" { these share the same 3-tuples ] [althougfh ED count is perhaps ~15 million]

Ive only looked at a few of these - but in general its usually possible to solve a B12347 with plus 3 clues in the 4-boxes. [ min is 2 and probably all can be solved in 4]

The ED count of the 4-boxes is ~ 10 million I believe bit I dont know how many actual 4-boxes are solvable with 2 [different] clues.

Code: Select all

+---+---+---+
|8..|...|675|
|...|...|928|
|...|...|341|
+---+---+---+
|...|9..|237|
|...|...|594|
|...|...|816|
+---+---+---+
|123|456|789|
|598|721|463|
|647|398|152|
+---+---+---+
             
+---+---+---+
|839|142|...|
|714|563|...|
|256|879|...|
+---+---+---+
|461|985|...|
|382|617|...|
|975|234|...|
+---+---+---+
|...|...|...|
|...|...|...|
|...|...|...|
+---+---+---+


For a specific 4 box - probably the maximum number of UAs that I can fit is 4 plus big one

Code: Select all

+---+---+---+                             
|12.|.78|...|                             
|3..|4..|...|                             
|4..|3..|...|                             
+---+---+---+                             
|215|..6|...|                             
|..6|..5|...|                             
|...|.87|...|                             
+---+---+---+                             
|...|...|...|                             
|...|...|...|                             
|...|...|...|                             
+---+---+---+  4 UA4s plus ? a big one     
                                           
+---+---+---+                             
|129|578|...|                             
|367|421|...|                             
|458|369|...|                             
+---+---+---+                             
|215|946|...|                             
|786|135|...|                             
|934|287|...|                             
+---+---+---+                             
|...|...|...|                             
|...|...|...|                             
|...|...|...|                             
+---+---+---+    so probably 5 clues needed



[coloin]

What are the lowest and the greatest number of cells that can be placed?
JPF

Hmmm i dont think mike's effort is going to be bettered ...

but I guess this might be the next one to look at

Code: Select all

+---+---+---+
|1..|...|.34|
|...|1..|567|
|...|...|.89|
+---+---+---+
|.1.|.34|...|
|...|567|...|
|...|89.|.2.|
+---+---+---+
|.34|...|...|
|79.|..2|...|
|865|...|..2|
+---+---+---+    705 sols


Is there is a grid with 1 solution - if so we can find the minum clues.....

[JPF]

Hmmm i dont think mike's effort is going to be bettered ...

Actually, it can...

Is there is a grid with 1 solution..

Code: Select all

+---+---+---+
|1..|...|.89|
|...|1..|6.5|
|...|...|.73|
+---+---+---+
|.1.|3..|...|
|...|4.7|...|
|...|658|.2.|
+---+---+---+
|4..|...|...|
|6.3|..2|...|
|.79|...|..2|
+---+---+---+    23C


Edit:

Code: Select all

+---+---+---+
|1..|...|..5|
|...|1..|7.3|
|...|...|.98|
+---+---+---+
|.1.|3..|...|
|...|4.7|...|
|...|869|.2.|
+---+---+---+
|3..|...|...|
|4.6|..2|...|
|.89|...|..2|
+---+---+---+   22C


JPF

[m_b_metcalf]

What are the lowest and the greatest number of cells that can be placed?

Here are two candidates for the lowest number of cells that can be placed:

Code: Select all

Common clues (51):
 . . . . 6 . 3 9 1
 . 1 . . . . 2 8 6
 . . . 1 . . 4 5 7
 1 . . . . . 5 3 4
 . . . . 1 . 6 2 9
 . . 9 . . . 1 7 8
 7 9 3 5 4 6 8 1 2
 5 8 6 7 2 1 9 4 3
 4 2 1 3 8 9 7 6 5  5 solutions, 2 placed

Common clues (52):
 . . . . . 2 8 1 4
 . 1 . . . . 2 5 9
 . . . 1 . . 6 3 7
 1 . . . . . 7 6 8
 . . . . 1 . 4 9 5
 . . 4 . . 8 3 2 1
 6 2 1 7 4 9 5 8 3
 7 5 8 3 2 1 9 4 6
 3 4 9 8 5 6 1 7 2  4 solutions, 3 placed

Hmmm i dont think mike's effort is going to be bettered ...

Actually, it can...

JPF

I haven't understood Are we still talking about B?

[JPF]

I was speaking about B having 4 solutions...

JPF

[m_b_metcalf]

Unless we're mistaken, I agree with Mike: the minimum number of solutions is 4.
The corollary question is now: How many cells can be placed?
In Mike's example, the number of placed cells is 16 (65-49)
[snip]
What are the lowest and the greatest number of cells that can be placed?

I suppose you mean for 4-solutions B grids. For many-solution grids I get:

Code: Select all

 . . . . . . 8 1 5
 . 1 . . . . 6 2 7
 . . . 1 . . 3 9 4
 1 . . . . . 9 5 2
 . . . . 1 . 4 7 8
 . . . . . . 1 3 6
 7 3 8 6 5 1 2 4 9
 5 4 9 8 3 2 7 6 1
 2 6 1 4 7 9 5 8 3  49 clues, many solutions, none placed

 9 5 . . 6 . 7 8 1
 8 1 6 5 9 7 2 4 3
 . 7 . 1 8 . 5 6 9
 1 9 . . 7 . 6 5 8
 . 8 5 . 1 6 4 9 7
 4 6 7 8 5 9 3 1 2
 5 3 1 6 2 8 9 7 4
 7 2 8 9 4 5 1 3 6
 6 4 9 7 3 1 8 2 5  70 clues, 4 solutions, 21 placed


I look forward to seeing your 2(?)-solutions B grid!

Mike